Theorem naryrcl 48578

Description: Reverse closure for n-ary (endo)functions. (Contributed by AV, 14-May-2024.)

Hypothesis

Ref	Expression
naryfval.i	⊢ 𝐼 = (0..^𝑁)

Assertion

Ref	Expression
naryrcl	⊢ (𝐹 ∈ (𝑁-aryF 𝑋) → (𝑁 ∈ ℕ0 ∧ 𝑋 ∈ V))

Proof of Theorem naryrcl

Dummy variables 𝑛 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-naryf 48574	. 2 ⊢ -aryF = (𝑥 ∈ ℕ0, 𝑛 ∈ V ↦ (𝑛 ↑m (𝑛 ↑m (0..^𝑥))))
2	1	elmpocl 7653	1 ⊢ (𝐹 ∈ (𝑁-aryF 𝑋) → (𝑁 ∈ ℕ0 ∧ 𝑋 ∈ V))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3464  (class class class)co 7410   ↑_m cmap 8845  0cc0 11134  ℕ₀cn0 12506  ..^cfzo 13676  -aryF cnaryf 48573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-xp 5665  df-dm 5669  df-iota 6489  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-naryf 48574

This theorem is referenced by:  naryfvalelfv  48579  0aryfvalelfv  48582  fv1arycl  48584  1arymaptfv  48587  fv2arycl  48595  2arymaptfv  48598